Can someone help? I'm just wondering what the connection is between a span (v1,v2) and independence in R3. How could that span be independent as well?

Question

phi · Answer

yes

empty · Answer

Yeah, definitely, suppose v1 and v2 are linearly dependent then that means they don't span 2D space, only a 1D line. Then there's no way for v3 to span the two dimensions that v1 and v2 don't have so then you've got a contradiction and v1, v2, and v3 don't span 3D space anymore

anonymous · Answer

Sorry, Linear Algebra confuses me so much. So basically, it's true because v1 and v2 only span a 1D line, which means that v3 does not exist because R3 spans a 2D line, right?

phi · Answer

linearly dependent means one of the vectors  (or two for that matter) are a linear combination of the remaining vectors.
For example, if v3 was *dependent*  then you know
v3 = A* v1 + B*v2  where A and B are not both zero

if both v2 and v3 are dependent, you would have
v3= A*v1
v2 = B*v1
and all 3 vectors v1,v2,v3 are scaled versions of the same vector.

if all 3 are independent, and you drop one of them, the remaining two are still independent (can not be a combination of the other one)

anonymous · Answer

Oh, okay! That is a lot clearer and it helps so much! I just have to think about it a bit more, but it really helps! Thank you.